Positive Solutions for Schrödinger-Poisson Systems with Sign-Changing Potential and Critical Growth

Abstract

<jats:p>In this paper, we study the existence of positive solutions for Schrödinger-Poisson systems with sign-changing potential and critical growth. By using the analytic techniques and variational method, the existence and multiplicity of positive solutions are obtained.</jats:p>

Positive Ground State Solutions for Schrödinger-Poisson System with General Nonlinearity and Critical Exponent

“…To the best of our knowledge, researchers only obtained a few results about the Schr ödinger-Poisson system with critical exponent on bounded domain, see for instance [1], [6][7][8][9][10][11].…”

“…In [6], assuming that η =λ, f (x,u)=λu q−1 , λ>0 and 1<q<2, via using the variational method, the authors proved that system (1.1) has at least two positive solutions and one of the solutions is a ground state solution for all λ∈(0,λ * ), where λ * is a positive constant. In [7], let η = −1, f (x,u) = λ f λ (x)u q−1 , f λ = λ f + + f − , λ > 0 and 1 < q < 2, by using the variational method and analytic techniques, they got that system (1.1) has at least two positive solutions and one of the solutions is a ground state solution for all λ ∈ (0,λ * ), where λ * is a positive constant. In [8], when η =−1, f (x,u)=λu q−1 , λ>0 and 2<q<6, by the Mountain pass theorem and the concentration compactness principle, they obtained that if 2 < q ≤ 4, system (1.1) has at least one positive ground state solution for all λ > λ * , where λ * is a positive constant; if 4 < q < 6, system (1.1) has at least one positive ground state solution for all λ > 0.…”